Vectors 3

Ok, so can someone please check my work for me? I am just not sure on one part, finding the angle of cos(theta). Btw this is the dot product.

a and b both have u-hats on them

6(a)^2 + 7(a)(b)-20(b)^2
=6(1)^2+7(0.5)-20(1)^2
=-10.5

dot product work:
a . b = |a||b|cos(theta)
=(1)(1)cos60
=0.5

I was wondering if I had my angle theta correct
To get it, I used the cos law,
so theta is the angle across from 1 in my diagram.. am I supposed to look for the angle between a and b or between a and a-b.. Because I looked for the one between a and a-b
cos(theta)=(1+5-1)/(2(1)(5))
theta=0.5

Ok, thanks! just one more question though, when you go from the 2nd step to the 3rd step, I'm wondering if there is a dot product rule or property that you are supposed to use to know that you have to seperate the squared into (a-b) (a-b), or, did you just know that you had to do that, in order too find a.b

I'm wondering if there is a dot product rule or property that you are supposed to use to know that you have to seperate the squared into (a-b) (a-b), or, did you just know that you had to do that, in order too find a.b

that and the step before that
(a-b)^2 = (a-b) . (a-b)
is there a rule or property of the dot product that says you need to or can split the squared into two, or do you just do that from normal mathematical approach?

also with
are these three too, rules or property of the dot product as well?